If the sum of the first 9 terms of an AP is equal to the sum of its first 11 terms, then find the sum of its first 20 terms.

Answer

To solve the problem step by step, we will use the formula for the sum of the first n terms of an arithmetic progression (AP).

### Step 1: Write the formula for the sum of the first n terms of an AP
The sum of the first n terms (S_n) of an AP is given by the formula:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
where \(a\) is the first term, \(d\) is the common difference, and \(n\) is the number of terms.

### Step 2: Set up the equation for the sum of the first 9 and 11 terms
According to the problem, the sum of the first 9 terms is equal to the sum of the first 11 terms:
\[ S_9 = S_{11} \]
Using the formula, we can express this as:
\[ \frac{9}{2} \times (2a + 8d) = \frac{11}{2} \times (2a + 10d) \]

### Step 3: Simplify the equation
We can eliminate \(\frac{1}{2}\) from both sides:
\[ 9(2a + 8d) = 11(2a + 10d) \]
Expanding both sides gives:
\[ 18a + 72d = 22a + 110d \]

### Step 4: Rearrange the equation
Rearranging the equation to isolate terms involving \(a\) and \(d\):
\[ 18a - 22a = 110d - 72d \]
This simplifies to:
\[ -4a = 38d \]
Dividing both sides by -2 gives:
\[ 2a + 19d = 0 \quad \text{(Equation 1)} \]

### Step 5: Find the sum of the first 20 terms
Now, we need to find the sum of the first 20 terms using the same formula:
\[ S_{20} = \frac{20}{2} \times (2a + 19d) \]
This simplifies to:
\[ S_{20} = 10 \times (2a + 19d) \]

### Step 6: Substitute Equation 1 into the sum formula
From Equation 1, we know that \(2a + 19d = 0\). Substituting this into our equation for \(S_{20}\):
\[ S_{20} = 10 \times 0 = 0 \]

### Conclusion
Thus, the sum of the first 20 terms of the AP is:
\[ \boxed{0} \]